Greg Allen snapped a 3-all tie with a two-run homer in the seventh inning and the Indians hung on to edge the Red Sox 5-4 at Fenway Park.
Michael Brantley's two-run shot in the sixth tied the game at three for Cleveland, which has won eight of its last nine games.
Melky Cabrera also homered in support of Corey Kluber's 16th win.
Cody Allen notched his 25th save.
The Indians and Red Sox play the second of their four-game series tonight at Fenway Park.
Cleveland is 72-and-52, 13 games ahead of the Twins for first in the AL Central.
Shane Bieber takes the mound for the Indians opposite Boston's Nathan Eovaldi.